= Solution
For the given <weighted inner product>, a <self-adjoint operator> $L$ obeys $L=L^\dagger$, including equality of its domain with the adjoint domain. In the differential-equation calculation this requires
$$
\langle f,Lg\rangle=\langle Lf,g\rangle
$$
for all functions in a specified common domain, with endpoint conditions that remove the boundary form. Equality of these inner products by itself describes symmetry; the domain conditions complete the self-adjoint realization.
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